General Computation of Entropy, Joint Entropy, Conditional Entropy, and Average Mutual Information

Information theory features several core concepts for quantifying information uncertainty and correlation, including entropy, joint entropy, conditional entropy, and average mutual information. These metrics are widely applied in machine learning and communication fields.

Entropy measures the uncertainty of a random variable, where higher values indicate greater uncertainty. Computation requires knowledge of the variable's probability distribution, achieved by summing the products of each possible outcome's probability and its logarithm. In code implementation, this typically involves iterating through probability values and applying the entropy formula H(X) = -Σ p(x)log p(x).

Joint entropy extends the entropy concept to measure the combined uncertainty of two or more random variables. Calculation requires knowledge of the joint probability distribution of these variables. Programming implementations often use nested loops to process multidimensional probability arrays and compute H(X,Y) = -ΣΣ p(x,y)log p(x,y).

Conditional entropy measures the remaining uncertainty of one random variable given knowledge of another variable. It reflects dependencies between variables and requires conditional probabilities for computation. Algorithmically, this involves first computing joint probabilities and marginal probabilities, then applying H(Y|X) = H(X,Y) - H(X).

Average mutual information quantifies the mutual dependence between two random variables, representing the reduction in uncertainty about one variable when the other is known. Computation requires comparing joint probabilities with the product of marginal probabilities. The key implementation uses I(X;Y) = ΣΣ p(x,y)log[p(x,y)/(p(x)p(y))], which can be optimized using vectorized operations in numerical computing libraries.

In practical applications, general computational procedures for these concepts typically begin with constructing probability distribution tables, followed by iterative summation according to respective formulas. For continuous variables, discretization processing may be necessary first, often implemented through binning or quantization algorithms before probability estimation.